Failure-Time Mixture Models: Yet Another Way to Establish Efficacy


We propose to use mixture survival models to establish the efficacy of the trial treatment. In particular, we consider the lognormal distribution to model the right-censored event time and a logistic regression for the incidence part of the model. The model attempts to estimate simultaneously the effects of treatments on the acceleration/deceleration of the timing of a given event and the surviving fraction-the proportion of the population for which the event may never occur. We use the SAS/IML subroutine NLPTR to obtain the maximum likelihood estimates of the model parameters. The estimates of the standard errors of the parameter estimates are computed from the inverse of the observed information matrix. We use the Cox-Snell residual plot based on the unconditional survivor function for evaluating goodness-of-fit of the model. The principal research hypothesis will be that under the trial treatment, the time-to-event will be more decelerated/accelerated compared to the control, given that the event occurs. We suggest that this methodology could be considered as a means to establish efficacy. We emphasize that there can be a substantial advantage to using mixture models even when the log-rank test is valid and significant. Data on overall survival time from a typical colorectal cancer clinical trial are used to illustrate the procedure.

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  1. 1.

    Hill BM. Bayesian nonparametric survival analysis: A comparison of the Kaplan-Meier and Berliner-Hill estimators. In JP Klein and PK Goel, eds. Survival Analysis: State of the Art. The Netherlands: Kluwer Academic Publishers, 1992, 25–46.

    Google Scholar 

  2. 2.

    Anderson PK, Borgan O, Gill RD, Keiding N. Statistical Models Based on Counting Processes. New York, NY: Springer-Verlag, 1993.

    Google Scholar 

  3. 3.

    Cox DR, Oakes D. Analysis of Survival Data. London: Chapman and Hall; 1984.

    Google Scholar 

  4. 4.

    Miller RG, Jr. What price Kaplan-Meier? Biometrics. 1983;39:1077–1081.

    Article  Google Scholar 

  5. 5.

    Miller RD, Jr. Survival Analysis. New York: John Wiley; 1981.

    Google Scholar 

  6. 6.

    Boag JW. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J Roy Stat Society. 1949;11:15–53.

    Google Scholar 

  7. 7.

    Farewell VT. A model for a binary variable with time-censored observations. Biometrics. 1977;64:1,43–46.

    Article  Google Scholar 

  8. 8.

    Pierce DA, Stewart WH, Kopecky KJ. Distribution-free regression analysis of grouped survival data. Biometrics. 1979;35:785–793.

    CAS  Article  Google Scholar 

  9. 9.

    Farewell VT. The use of mixture models for the analysis of survival data with long-term survivors. Biometrics. 1982;38:1041–1046.

    CAS  Article  Google Scholar 

  10. 10.

    Taylor JMG, Kim DK. Statistical models for analyzing time-to-occurrence data in radiobiology and radiation oncology. Int J Radiation Biology. 1993;52:459–468.

    Google Scholar 

  11. 11.

    Taylor JMG. Semi-parametric estimation in failure time mixed models. Biometrics. 1995;51:899–907.

    CAS  Article  Google Scholar 

  12. 12.

    Yamaguchi K. Accelerated failure-time regression models with a regression model of surviving fraction: An application to the analysis of permanent employment in Japan. J Am Stat Assoc. 1992;87:284–292.

    Google Scholar 

  13. 13.

    Peng Y, Dear KBG, Denham JW. A generalized F mixture model for cure rate estimation. Stat Med. 1998;17:813–830.

    CAS  Article  Google Scholar 

  14. 14.

    Sy JP, Taylor JMG. Estimation in a Cox proportional hazards cure model. Biometrics. 2000;56:227–236.

    CAS  Article  Google Scholar 

  15. 15.

    SAS Institute Inc. SAS/STAT User’s Guide. Version 6, Fourth Edition, Volume 2. Cary, NC: SAS Institute, Inc.; 1994.

    Google Scholar 

  16. 16.

    SAS Institute Inc. SAS/IML Software: Changes and Enhancements, 6.11. Cary, NC: SAS Institute Inc.; 1995.

    Google Scholar 

  17. 17.

    Escobar LA, Meeker WQ. Fisher information matrices with censoring, truncation, and explanatory variables. Statistica Sinica. 1998;8:221–237.

    Google Scholar 

  18. 18.

    Collett D. Modeling Survival Data in Medical Research. London: Chapman Hall; 1994.

    Google Scholar 

  19. 19.

    Allison PD. Survival Analysis Using the SAS System: A Practical Guide. Cary, NC: SAS Institute Inc.; 1995.

    Google Scholar 

  20. 20.

    Geiser PW, Chang MN, Rao PV, Shuster JJ, Pullen J. Modeling cure rates using the Gompertz model with covariate information. Stat Med. 1998:17:831–839.

    Article  Google Scholar 

  21. 21.

    Lehmann EL. Elements of Large Sample Theory. New York: Springer-Verlag; 1999.

    Google Scholar 

  22. 22.

    Farewell VT. Mixture models in survival analysis: Are they worth the risk? Canadian J Stat. 1986;14(3):257–262.

    Article  Google Scholar 

  23. 23.

    Ghitany ME, Mailer RA, Zhou S. Exponential mixture models with long-term survivors and covariates. J Multivariate Analysis. 1994;49:218–241.

    Article  Google Scholar 

  24. 24.

    Koti KM. Mixture models combining logistic regression with failure-time parametric regression: An application to NASA/JSC DCSTIME data. Proceedings of the Biometrics Section of the American Statistical Association. 1998;198–203.

    Google Scholar 

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Correspondence to Kallappa M. Koti PhD.

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Koti, K.M. Failure-Time Mixture Models: Yet Another Way to Establish Efficacy. Ther Innov Regul Sci 35, 1253–1260 (2001).

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Key Words

  • Clinical trial
  • Accelerated failure-time
  • Surviving fraction
  • Wald test